Some Problems of Definability in Recursive Function Theory

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Publisher Summary This chapter discusses several problems of definability in recursive function theory. It presents several of the basic facts of elementary recursive function theory and describes several structures that have been principal objects of study in recent development of the theory. The chapter considers the notion of definability and provides a brief discussion of the open problems. The notion of recursive function is the central concept in the theory; the partial recursive functions play a fundamental technical role. They can be effectively enumerated; which means that an effective procedure exists for listing all the sets of instructions for all the partial recursive functions. The chapter describes the concepts of recursive function and partial recursive function. A few basic theorems in the elementary theory are also presented. The first theorem gives a fundamental relationship among the concepts of recursive function. The second theorem gives a technically useful characterization of recursive enumerability. The third theorem simplifies several concerns of later parts of the theory.